Here are the functions and operations as shown in the image:

Given functions:

• $f(x) = 6x^2 - 35x + 49$
• $g(x) = \frac{4x}{3} + 2$
• $j(x) = \frac{2}{x} - 1$
• $k(x) = 3x - 7$
• $m(x) = 4x + 22$
• $h(x) = x^2 + 3x - 4$

Tasks:

1. $(h - k)(x)$
2. $\left(\frac{g}{j}\right)(x)$
3. $(f \cdot k)(x)$
4. $(k + m)(x)$
5. $h(k) - (k \cdot m)(x)$

Let's solve them step by step:

1. $(h - k)(x)$

This represents the difference between the functions $h(x)$ and $k(x)$: $(h - k)(x) = h(x) - k(x) = (x^2 + 3x - 4) - (3x - 7)$ Simplify the expression: $= x^2 + 3x - 4 - 3x + 7 = x^2 + 3x - 3x - 4 + 7 = x^2 + 3$ So, $(h - k)(x) = x^2 + 3$.

2. $\left(\frac{g}{j}\right)(x)$

This represents the division of the functions $g(x)$ and $j(x)$: $\left(\frac{g}{j}\right)(x) = \frac{g(x)}{j(x)} = \frac{\frac{4x}{3} + 2}{\frac{2}{x} - 1}$ To simplify this, multiply the numerator and denominator by $x$: $\frac{\left(\frac{4x}{3} + 2\right) \cdot x}{\left(\frac{2}{x} - 1\right) \cdot x} = \frac{\frac{4x^2}{3} + 2x}{2 - x}$ So, $\left(\frac{g}{j}\right)(x) = \frac{\frac{4x^2}{3} + 2x}{2 - x}$.

3. $(f \cdot k)(x)$

This represents the product of the functions $f(x)$ and $k(x)$: $(f \cdot k)(x) = f(x) \cdot k(x) = (6x^2 - 35x + 49)(3x - 7)$ Expanding this: $(6x^2 - 35x + 49)(3x - 7) = 6x^2(3x) + 6x^2(-7) - 35x(3x) - 35x(-7) + 49(3x) + 49(-7)$ $= 18x^3 - 42x^2 - 105x^2 + 245x + 147x - 343$ Combine like terms: $= 18x^3 - 147x^2 + 392x - 343$ So, $(f \cdot k)(x) = 18x^3 - 147x^2 + 392x - 343$.

4. $(k + m)(x)$

This represents the sum of the functions $k(x)$ and $m(x)$: $(k + m)(x) = k(x) + m(x) = (3x - 7) + (4x + 22)$ Simplify the expression: $= 3x - 7 + 4x + 22 = 7x + 15$ So, $(k + m)(x) = 7x + 15$.

5. $h(k) - (k \cdot m)(x)$

This is a two-part problem:

1. First, evaluate $h(k(x))$, which means substituting $k(x) = 3x - 7$ into $h(x)$: $h(k(x)) = h(3x - 7) = (3x - 7)^2 + 3(3x - 7) - 4$ Expanding the square and simplifying: [ = (9x^2 - 42x + 49) + (9x - 21) - 4 = 9x^2 - 42x + 49 +